THE INCIDENCE CHROMATIC NUMBER OF SOME GRAPH
LIU XIKUI AND LI YAN
Received 1 April 2003 and in revised form 5 December 2003
The concept of the incidence chromatic number of a graph was introduced by Brualdi and Massey (1993). They conjectured that every graph G can be incidence colored with ∆(G) + 2 colors. In this paper, we calculate the incidence chromatic numbers of the com- plete k-partite graphs and give the incidence chromatic number of three infinite families of graphs.
1. Introduction
Throughout the paper, all graphs dealt with are finite, simple, undirected, and loopless. Let G be a graph, and let V(G), E(G), ∆(G), respectively, denote vertex set, edge set, and maximum degree of G. In 1993, Brualdi and Massey [3] introduced the concept of incidence coloring. The order of G is the cardinality |v(G)|. The size of G is the cardinality |E(G)|.Let
I(G) = (v,e) | v ∈ V, e ∈ E, v is incident with e (1.1) be the set of incidences of G. We say that two incidences (v,e)and(w, f )areadjacent provided one of the following holds: (i) v = w; (ii) e = f ; (iii) the edge vw = e or vw = f . Figure 1.1 shows three cases of two incidences being adjacent. An incidence coloring σ of G is a mapping from I(G)toasetC such that no two adjacent incidences of G have the same image. If σ : I(G) → C is an incidence coloring of G and |C|=k, k is a positive integer, then we say that G is k-incidence colorable.
Copyright © 2005 Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences 2005:5 (2005) 803–813 DOI: 10.1155/IJMMS.2005.803 804 The incidence chromatic number of some graph
v v v ∗∗ ∗ ∗ efe e f
w ∗ w ∗ v = w e = f vw = e
(a) (b) (c)
Figure 1.1. Cases of two incidences being adjacent.
The minimum cardinality of C for which there exists an incidence coloring σ : I(G) → C is called the incidence chromatic number of G, and is denoted by inc(G). A partition {I1,I2,...,Ik} of I(G) is called an independence partition of I(G)ifeachIi is independent in I(G) (i.e., no two incidences of Ii are adjacent in I(G)). Clearly, for k ≥ inc(G), G is k-incidence colorable. We may consider G as a digraph by splitting each edge uv into two opposite arcs (u,v) and (v,u). Let e = uv.Weidentify(u,e)withthearc(u,v). So I(G) may be identified with the set of all arcs A(G). Two distinct arcs (incidences) (u,v)and(x, y)areadjacentifone of the following holds (see Figure 1.2): (1’) u = x; (2’) u = y and v = x; (3’) v = x. This concept was first developed by Brualdi and Massey [3] in 1993. They posed the incidence coloring conjecture (ICC), which states that for every graph G,inc(G) ≤ ∆ + 2. In 1997, Guiduli [5] showed that incidence coloring is a special case of directed star arboricity, introduced by Algor and Alon [1]. They pointed out that the ICC was solved in the negative following an example in [1]. Following the analysis in [1], they showed that inc(G) ≥ ∆ + Ω(log∆), where Ω = 1/8 −◦(1). Making use of a tight up- per bound for directed star arboricity, they obtained the upper bound inc(G) ≤ ∆ + O(log∆). Brualdi and Massey determined the incidence chromatic numbers of trees, complete graphs, and complete bipartite graphs [3]; Chen, Liu, and Wang determined the incidence chromatic numbers of paths, cycles, fans, wheels, adding-edge wheels, and complete 3- partite graphs [4]. In this paper, we will consider the incidence chromatic number for complete k-partite graphs. We will give the incidence chromatic number of complete k-partite graphs, and also give the incidence chromatic number of three infinite families of graphs. Let k be positive integer, put [k] = 1,2,...,k. We state first the following definitions. Definition 1.1. For a graph G(V,E)withvertexsetV and edge set E, the incidence graph I(G)ofG is defined as the graph with vertex set V(I(G)) and edge set E(I(G)). Definitions not given here may be found in [2]. L. Xikui and L. Yan 805
u(x) v(y) v(x)
∗∗ ∗ ∗ e
∗ ∗ vyu(x) u y
(a) (b) (c)
Figure 1.2. Cases of two arcs (incidences) are adjacent.
2. Some useful lemmas and properties of incidence chromatic number Lemma 2.1. Let T beatreeofordern ≥ 2 with maximum degree ∆. Then inc(T) = ∆ +1. Lemma 2.2. AgraphG is k-incidence colorable if and only if its incidence graph I(G) is k-vertex colorable, that is, inc(G) = χ(I(G)). Let M ={(ue,ve) | e = uv ∈ E(G), (ue,ve) ∈ E(I(G))},thenM forms a perfect match- ing of incidence graph I(G). The following lemmas are obvious. Lemma 2.3. The incidence graph I(G) of a graph G is a graph with a perfect matching.
Lemma 2.4. For a graph G, v ∈ V(G),letBv ={(u,uv) | uv ∈ E(G), u ∈ V(G)}, Av = {(v,vu)|uv ∈ E(G), u ∈ V(G)}, then {Bv} is an independence-partition of incidence graph I(G), and the induced subgraph G[Av] of I(G) is a clique graph. By the definition of incidence graph, it is easy to give the proof. Lemma 2.5. Let ∆ be the maximum degree of graph G, I(G) the incidence graph, then com- plete graph K∆+1 is a subgraph of I(G).
Proof. Let d(u) = ∆, p = ∆,andek = uv1, e2 = uv2, ..., ep = uv1 be the edges of G. p incidences in Iu ={(u,e1),(u,e2),...,(u,ep)} are adjacent to each other. For an incidence in Iv ={(vi,viu) | uvi ∈ G,1≤ i ≤ p},(vi,viu) is adjacent to all incidences in Iu.Sincep + 1incidences(u,e2),...,(u,ep),(vi,viu)areverticesofI(G), by the definition of incidence graph, we can complete the proof. Lemma 2.6. For a simple graph G with order n, inc(G) = n = ∆(G)+1, when ∆(G) = n − 1.
Proof. Let |V(G)|=∆ +1= v(G), by Lemma 2.4, {Bv} is an independence-partition of incidence graph I(G), then χ(I(G)) ≤ v(G). By Lemma 2.5, K∆+1 is the subgraph of I(G), thus χ(I(G)) ≥ v(G), then inc(G) = χ(I(G)) = ∆ +1,asrequired. The following corollaries can be easily verified. Corollary 2.7. Let G be a graph with order n (n ≥ 2), then inc(G) ≤ ∆ +2, when ∆(G) = n − 2. 806 The incidence chromatic number of some graph
In fact, for graphs G with order n, we can give each incidence in I(G)properinci- dence coloring as follows. Let V(G) ={v1,v2,...,vn} be the vertex set and C ={1,2,...,n} the color set. For i, j = 1,2,...,n,weletσ(vi,vivj ) = j. It is easy to see that the coloring above is an incidence coloring of G only with n colors. That is, inc(G) ≤ ∆ +2,when ∆ ≥ n − 2.
Corollary 2.8. Let Wn be the wheel graph with order n +1. Then inc(Wn) = n +1. Lemma 2.9. Let H beasubgraphofG, then inc(H) ≤ G.
Lemma 2.10. Let G be union of disjoint graphs G1,G2,..., and Gt.IfGi has an m-incidence coloring for all i = 1,2,...,t, then G has an m-incidence coloring. That is inc(G) = max{inc(Gi) | i = 1,2,...,t}.
Proof. To prove this lemma, we only need to prove that G1 ∪ G2 has an m-incidence {I I ... I } I G {I I ... I } coloring. Let 1, 2, , m be an independence partition of ( 1), and 1, 2, , m I G {I ∪ I I ∪ I ... I ∪ I } an independence partition of ( 2). Then 1 1, 2 2, , m m forms an independence-partition of I(G1) ∪ I(G2). Hence G has an m-incidence coloring. The proof of the lemma is complete. Theorem 2.11. Let G be a graph with maximum degree ∆(G)=n−2 and minimum degree δ(G)≤[n/2]−1, then inc(G)=n−1=∆(G)+1.
Proof. Let V(G) ={v1,v2,...,vn}, d(v1) = δ(G), and u/∈ V(G). Consider the auxiliary graph G with vertex set V(G ) = V(G) ∪{u} and edge set E(G ) = E(G) ∪{uvi | i = 2,3,...,n}. It follows that ∆(G ) = n − 1. Let G = G −{v1},then∆(G ) = n − 1, by Lemma 2.5,inc(G) = n.ForcolorsetC ={1,2,...,n}, suppose that σ is the n-incidence coloring of G with color set C. Without loss of generality, let σ (vi,vivj ) = j (vivj ∈ E(G)) and σ (vi,viu) = 1(i = 2,3,...,n), σ (u,uvi) = i (i = 2,3,...,n). In incidence set I(G), incidences (vi,vivj)(i, j = 2,3,...,n,andi = j)arealladjacentto(vi,viu)and (vj ,vj u), thus the color n cannot be used to color any incidence in I(G −{u}). De- N v ={v v ... v } v σ note by ( 1) i1 , i2 , , iδ the vertices adjacent to 1. The incidence coloring of graph G may be extended to an incidence coloring σ of graph G.Forx, y ∈ V(G)and x y/∈{v }∪N v σ x xy = σ x xy ∆ G = n − v k = , 1 ( 1), let ( , ) ( , ). Because ( ) 2, for vertex ik ( ... δ v ∈ V G v v ∈/ E G σ v v v = t 1,2, , ), there exists a vertex tk ( )suchthat ik tk ( ). Let ( ik , ik 1) k.At v v v ∈ I G k = ... δ last,wegiveincidences( 1, 1, ik ) ( )( 1,2, , ) the color used to color incidence (u,uvi) ∈ I(G )(i = 2,3,...,n). Since d(v1) = δ ≤ [n/2] − 1, then 2d(v1) ≤ 2[n/2] − 2 ≤ n − 2, that is, d(v1) ≤ n − 2 − d(v1), thus we can select d(v1) colors to incidence color, thus σ is a proper n-incidence coloring of G.Theproofiscompleted. For the general case, using the way similar to Theorem 2.11,wecangiveastronger result. Theorem 2.12. For graph G with order n and maximum degree ∆(G) = n − k, inc(G) = n − k +1= ∆(G)+1,whenminimumdegreeδ(G) ≤ [(n − k +2)/2] − 1.
For a graph G, if there exists two vertices u,v ∈ V(G)\v1 such that d(u) = n − 3, d(v) ≤ n − 4, and uv∈ / E(G), we say that G is with the property P. L. Xikui and L. Yan 807
Theorem 2.13. For graph G with order n and maximum degree ∆(G) = n − 3, inc(G) ≤ ∆(G)+2(n ≥ 4), when minimum degree δ(G) ≤ [n/2] − 1.
Proof. By Vn ={v1,v2,...,vn} we denote a labeling of the vertices of G and let d(v1) = δ(G). For n = 4,5, the desired result follows from Lemma 2.1. For the case n ≥ 6, the proof can be divided into two cases. Case 1. G is with the property P. Consider the auxiliary graph G = G + uv.Since∆(G) = n−2andδ(G) ≤ [n/2]−1, by Theorem 2.11,inc(G) = n−1 = ∆(G)+1. Thus inc(G) ≤ inc(G) = ∆(G)+2. Case 2. G not with the property P.Fortwoverticesu,v ∈ V(G)\v1,letV1(G) ={v ∈ V(G) | dG(v) = n − 3} and V2(G) = V(G)\{V1(G) ∪{v1}}. Subcase 1. V2(G) =∅.Letw/∈ V(G)andG = G + w + {wv | v ∈ V1(G)},then∆(G ) = n − 1andδ(G) ≤ [n/2] − 1. By Theorem 2.11, using similar methods as in the proof of Theorem 2.11, we can prove the desired result inc(G) ≤ n − 1. Subcase 2. V2(G) =∅.Letx be the arbitrary vertex in V1(G), then N(x) = V2(G). For arbitrary vertex v ∈ V2(G), since d(v) ≤ n − 4, then |V2(G)|≥3, and there exists two ver- u v V G u v ∈/ E G G = G u v G P tices 1, 1 in 2( )suchthat 1 1 ( ). Let 1 + 1 1.If 1 is with the property , G ≤ G ≤ n − V G ={v ∈ V G | d v = n − } then inc( ) inc( 1) 1. Otherwise let 1( 1) ( 1) G1 ( ) 3 and V2(G1) = V(G1)\{V1(G1) ∪{v1}}.IfV2(G1) =∅,theninc(G1) ≤ ∆(G1)+2.IfV2(G1) = ∅ |V G |≥ u v V G u v ∈/ E G ,then 2( 1) 3; there exists two vertices 1, 1 in 2( 1)suchthat 2 1 ( 1). G = G u v G P |V G |≥ V G =∅ Let 2 1 + 2 2.If 2 is not with the property ,then 2( 2) 3when 2( 2) . We can also construct graph G3 that is not with the property P.Inthatway,wecanobtain G G G ... G ... P aserialofgraphs , 1, 2, , k, such that all the graphs are not with the property and |V2(Gk)|≥3. Let D(G) = v∈G d(v), then D(G) ≤ D(G1) ≤ D(G2) ≤···≤D(Gk) ≤ ···. G G |V G |= Because is the finite graph, there exists a graph k0 such that 2( k0 ) 3. Sup- V Gk ={u u u } v ∈ V Gk V Gk = N v dG u = pose that 2( 0 ) 1, 2, 3 and 1( 0 ), then 2( 0 ) ( ). Thus k0 ( 1) dG u = dG u = n − u u u k0 ( 2) k0 ( 3) 4, and 1, 2, 3 are without edge and adjacent to each other. Let G = G u u u u ∈ G \v d u =n − d u ≤n − u u ∈/ E G G k0 + 1 2,then 1, 3 1, ( 1) 3, ( 3) 4, and 1 3 ( ), then P G ≤ G ≤ G ≤···≤ G ≤ G ≤ n − is with the property ,thusinc( ) inc( 1) inc( 2) inc( k0 ) inc( ) 1. The proof is complete.
Theorem 2.14. Let u,v ∈ V(G) such that uv∈ / E(G) and NG(u) = NG(v), then inc(G) ≥ ∆ +2. Proof. The proof is by contradiction. Suppose that the graph G has an (∆ + 1)-incidence coloring with color set C ={1,2,...,∆ +1}.LetNG(u) ={x1,x2,...,x∆} and NG(v) ={y1, y2,..., y∆}. Then each of the incidences (xi,xiu)(1≤ i ≤ ∆) is colored the same, as are the incidences (yi, yiv). Without loss of generality, suppose k the color that (yi, yiv)has.Be- cause NG(u) = NG(v)and(u,x1x1)isadjacentto(y1, y1v), then (u,ux1) has a color other than k. Because (u,ux2)isadjacentto(y2, y2v),...,(u,ux∆)whichisadjacentto(y∆, y∆v), then (u,ux2),...,(u,ux∆) also has a color other than k, respectively. Further, the ∆ inci- dences (u,uxi)(1≤ i ≤ ∆)havedifferent colors, so the color k is different from that of incidences (u,uxi). On the other hand, (y1, y1v)and(x1,x1u) are neighborly incidences, so the color k is different from that of (x1,x1u). Thus k/∈ C, this gives a contradiction! Hence inc(G) ≥ ∆ +2. 808 The incidence chromatic number of some graph
3. The incidence chromatic number of complete k-partite graph G = K k k ≥ Theorem 3.1. Let n1,n2,...,nk be a complete -partite graph ( 2). Then
∆ +1, ∆(G) = n − 1, inc(G) = (3.1) ∆(G)+2, otherwise.
Proof. Let V(G) = V1 ∪ V2 ∪···∪Vk and |Vi|=ni (i = 1,2,...,k). Vi is the i-part ver- i i i tex set and Vi ={v ,v ,...,vn } (i = 1,2,...,k). Without loss of generality, we let n ≥ 1 2 i 1 n ≥ ··· ≥ n ∆ G = k−1 n 2 k.Thus ( ) m=1 m. The proof can be divided into the following two cases. i ∈{ ... k} n = G V G = Case 3. There exists 1,2, , such that i 1. We let the vertex set of be ( ) {v v ... v } m = k n i, 2, , m ,where i=1 i.ByLemma 2.6, it easy to draw the conclusion. Case 4. ni ≥ 2(1≤ i ≤ k). To complete the proof, we give an incidence coloring just with ∆ +2colorsfirstly. For j,t = 1,2,...,k, i = 1,2,...,nj ,ands = 1,2,...,nt,welet
t −1 nm s i = s t To complete the proof, it suffices to prove that G cannot be colored with ∆ +1 colors. It is obvious that each of the vertices in V1 is the maximum-degree vertex. For n1 ≥ 2, let u,v ∈ V1,thenuv∈ / E(G)andN(u) = N(v). Hence inc(G) ≥ ∆ +2 followsfrom Theorem 2.14. Therefore inc(G) = ∆ + 2, and the proof is completed. By Theorem 3.1, it is easy to obtain the theorem in [3, 4]. In fact, the incidence coloring σ given to determine the incidence chromatic number for complete 3-partite graphs is a special case of the coloring above. Hence, we obtain some corollaries as follows. Corollary 3.2. Let Kn be complete graph. Then inc(Kn) = n. The incidence coloring of K3,4 and K5 is given in Figure 3.1. 4. Incidence chromatic number of three families of graphs The planar graph Qn, which is called triangular prism, is defined by Qn = G(V(G),E(G)), where the vertex set V(G) = u1,u2,...,un ∪ v1,v2,...,vn, and the edges set E(Qn) consists L. Xikui and L. Yan 809 v2 1 3 u u u 1 2 3 4 5 2 2 3 4 1 4 1 4 v v 5 2 5 3 2 5 1 3 1 3 5 4 4 5 1 6 2 6 2 3 316123 12 2 3 3 1 54 v v v v 1 2 3 4 v4 v5 K 3,4 K5 (a) (b) Figure 3.1. An incidence coloring of K3,4 and K5,respectively. of two n-cycles u1,u2,...,un and v1,v2,...,vn,and2n edges (ui,vi),(ui,vi+1)foralli ∈ [n](v1 = vn+1). Theorem 4.1. For any integer n ≥ 3, ∆ +1= 5, n = 0mod(5), Q = inc n (4.1) ∆ +2= 6, otherwise. Proof. Because ∆(G) = 4, we know that inc(Qn) ≥ ∆ +1= 5. When n = 5k, k ≥ 1, we ∗ give a 5-incidence coloring σ of Q5k.Fori = 1,2,...,5k,let(ui,uiui ) be the incidence set {ui,uiw | w = vi+1,ui±1,vi}.Let ∗ σ ui,uiui = 1+2(i − 1)(mod5), 2 + 2(i − 1)(mod5), 3+2(i − 1)(mod5), 4 + 2(i − 1)(mod5) , (4.2) σ vi±1,vi±1vi = σ ui,uivi , σ w,wui = σ uimp1,ui∓1ui , σ vi,viui = σ ui+1ui+1vi+1 . It is easy to see that the coloring above is a proper 5-incidence coloring of Qn.Thus, we can only consider the case n = 5k.WewillfirstprovethatQn is 6-incidence colorable by explicitly giving a 6-incidence coloring σ of Qn for any integer n ≥ 3. At last, we will give the proof that Qn cannot be incidence coloring just with colors 1,2,3,4,5. The proof can be divided into the following three cases. Case 5. n = 3k (k ≥ 1). Let i = 3s + t (t ≤ 2), i = 1,2,...,n,thenQn has an incidence color- ing using 6 colors from the color set C ={1,2,...,n + r +1},asfollows:fori = 1,2,...,n, 810 The incidence chromatic number of some graph let t, t = 0, σ v v v = σ u u u = i, i i+1 i, i i+1 3, t = 0, σ vi,vivi−1 = σ ui,uiui−1 = t +1, (4.3) σ ui,uivi+1 = σ vi,viui−1 = σ ui,uiui+1 +3, σ ui,uivi = σ vi+1,vi+1ui+1 = σ ui+1,ui+1ui +3. Case 6. n = 3k +1(k ≥ 1). Let i = 3s + t(t ≤ 2). For i = 1,2,...,n,let 3, t = 0, 3, t = 0, 4, i = 1, σ v v v = σ u u u = i = i, i i+1 i, i i+1 6, 1, i = 5, 2, t, otherwise, t, otherwise, i = i = 6, 1, 5, 1, (4.4) σ vi,vivi− = 2, i = 2, σ ui,uiui− = 4, i = n, 1 1 t + 1, otherwise, t + 1, otherwise, σ ui ,ui ui +3, i = 1,n, σ u u v = σ v v u = +1 +1 i, i i i+1, i+1 i+1 5, i = n, σ u1u1v1 = σ v2v2v1 = 2, σ v1v1u3 = 3. Case 7. n = 3k +2(k ≥ 1). Let i = 3s + t (t ≤ 2), for i = 1,2,...,n,andwn+1 = w1, w = u,v; w0 = wn, w = u,v.Welet 3, t = 0, 5, i = n, σ u u u = i, i i+1 i = 6, 1, t, otherwise, σ ui ,ui ui +3, i = 1,n, +1 +1 σ u u v = σ v v u = i = i, i i i+1, i+1 i+1 3, 1, 5, i = n, L. Xikui and L. Yan 811 2, i = n, 3, t = 0, σ vi,vivi = +1 i = 4, 1, t, otherwise, 2, i = 1, σ ui,uivi = σ ui,uiui +3= 5, i = n − 1, +1 σ ui,uiui+1 +3, i = 1,n, 1, i = 1, σ ui,uiui−1 = σ vi,vivi−1 = t + 1, otherwise, σ ui,uiui +3, i = 1,n +1 σ v v u = i = n i, i i−1 2, , 6, i = 1, σ ui,uiui +3, i = 1,n, +1 σ u u v = i = n i, i i+1 2, , 4, i = 1, σ ui,uiui +3, i = 1,n, +1 σ v v u = i = n i, i i−1 2, , (4.5) 6, i = 1. It is easy to show that Qn is 6-incidence colorable. To complete the proof, it remains to be shown that there do not exist an incidence coloring using only 5 colors. Assume, on the contrary, that Qn is 5-incident colorable. For each vertex vi ∈ Qn, d(vi) = ∆(Qn). Thus, four incidences (ui,uivi), (ui−1,ui−1vi), (vi±1,vi±1vi) have the same color, without loss of generality, 1. For i = 1,2,...,n, the case is the same. Because there are 5 colors that can be used in incidence coloring, and the degree of each vertex vi in cycle v1v2 ···vnv1 is 4, thus the two incidences (vi,vivi+1)and(vi+4,vi+4vi+5)(or(vi−4,vi−4vi−5)) have the same color. If n = 5k, form the proof above, it is easy to obtain a contradiction. Thus, we have completed the prove. Theorem 4.2. Let G be a Hamilton graph with order n ≥ 3 and degree ∆ ≤ 3. Then inc(G) ≤ ∆ +2. Proof. When ∆ ≤ 2, by Lemma 2.2,inc(G) ≤ ∆ +2.When∆ = 3, by Lemma 2.3,wecan only consider the case d(v) = 3(∀v ∈ V(G)). Let {v1,v2,...,vn,v1} be the Hamilton cycle and S = E(G) \{vivi+1 | 1 ≤ i ≤ n − 1}. The proof can be divided into the following three cases. Case 8. n = 0mod(3). For i = 1,2,...,n,weletσ(vi,vivi+1) = 2i − 1(mod3) and σ(vi+1, vi+1vi) = 2i(mod3), where vn+1 = v1. Because the edges e ∈ S form a matching, thus we can incidence color the incidence uncolored with two new colors 3,4. Then, we have given G an incidence coloring with colors 0,1,...,4. 812 The incidence chromatic number of some graph n = v ∈ A j = n v ∈ A n = n − i = ... n Case 9. 0mod(3). Let j v1 ( 1, )and k vn ( 1, 1). For 1,2, , and vn+1 = v1,welet 2i − 1(mod3), i = 1, j, σ v v v = i = j = k i, i i+1 4, +1, 3, otherwise, 2i(mod3), i = 1, j − 1, σ v v v = i = j − = k i+1, i+1 i 3, 1 , (4.6) 4, otherwise, 1, n = 1mod(3)and j = 1mod(3), σ v v v = n = j = j , j 1 0, 2mod(3)and 0mod(3), 2, otherwise, σ v1,v1vj = n − 1mod(3). Since the edges e ∈ S \{v1vk} form a matching, thus we can incidence color the inci- dence uncolored with two new colors 3,4. Thus, we have given G an incidence coloring with colors 0,1,...,4. The plane check graph Cm,n is defined by V(Cm,n) ={vi,j | i ∈ [m]; j ∈ [n]}; E(Cm,n) = {vi,jvi,j+1 | i ∈ [m]; j ∈ [n − 1]}∪{vi,jvi+1,j | i ∈ [m − 1]; j ∈ [n]}, which is the Cartesian product of path Pm and Pn, Theorem 4.3. For plane graph Cm,n,wehaveinc(Cm,n) = 5. Proof. ∆(Cm,n) = 4, then inc(Cm,n) ≥ 5. We now give a 5-incidence coloring σ of Cm,n as follows: (i ∈ [m]; j ∈ [n]) σ vi,j,vi,j vi,j+1 = j +3(i − 1) mod(5) (j = n), σ vi,j+1,vi,j+1vi,j = j +4(i − 1) mod(5) (j = n), (4.7) σ vi,j,vi,j vi+1,j = j +2(i − 1) mod(5) (i = m), σ vi+1,j,vi+1,j vi,j = j +4(i − 1) mod(5) (i = m). It is easy to see that the coloring above is an incidence coloring of Cm,n.Thus inc(Cm,n) = 5. Remark 4.4. It is difficult to obtain the incidence chromatic number for some graphs. We have presented a hybrid genetic algorithm for the incidence coloring on graphs in [6]. The experimental results indicate that a hybrid genetic algorithm can obtain solutions of excellent quality of problem instances with different size. Acknowledgment This work is supported by China National Science Foundation. L. Xikui and L. Yan 813 References [1] I.AlgorandN.Alon,The star arboricity of graphs, Discrete Math. 75 (1989), no. 1–3, 11–22. [2] J.A.BondyandU.S.R.Murty,Graph Theory with Applications, American Elsevier Publishing, New York, 1976. [3] R.A.BrualdiandJ.J.Q.Massey,Incidence and strong edge colorings of graphs, Discrete Math. 122 (1993), no. 1-3, 51–58. [4] D.L.Chen,X.K.Liu,andS.D.Wang,The incidence chromatic number and the incidence coloring conjecture of graph, Mathematics in Economics 15 (1998), no. 3, 47–51. [5] B. Guiduli, On incidence coloring and star arboricity of graphs, Discrete Math. 163 (1997), no. 1– 3, 275–278. [6] X.K.LiuandY.Li,Algorithm for graph incidence coloring base on hybrid genetic algorithm, Chinese J. Engrg. Math. 21 (2004), no. 1, 41–47. 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